ANALYTIC SOLUTION OF RICCATI EQUATIONS OCCURRING IN OPEN-LOOP NASH MULTIPLAYER DIFFERENTIAL GAMES

In this paper we present explicit analytic solutions of coupled Riccati matrix differential systems appearing in open-loop Nash games. Two different cases are considered. Firstly, by means of appropriate algebraic transformations the problem is decoupled so that an explicit solution of the problem is available. The second is based on the existence of a solution of a rectangular Riccati type algebraic matrix equation associated with the problem.

1. INTRODUCTION.When noncooperative problems are tackled, a game theoretic approach is necessary: each control agent (decision maker or player) tries to optimize his own cost function which conflicts more or less with others.An equilibrium solution must be sought, and the Nash strategy is a natural choice.In this case, a player cannot improve his payoff by deviating unilaterally from his Nash strategy.Due to this noncooperation, the optimization problem of various players are strongly coupled and necessary conditions for open-loop Nash strategy lead to complex two-point boundary value problems.
Consider a N-players linear quadratic differential game defined by N x'= Ax + Biui; x(0)-x 0 (1.1) i=l with the cost functionals associated with the players 1 T x!+ (xTOix/ uRijuj)d Ji " x.f Kiy 0 j=l x(tl where all matrices are n x n symmetric with Rii for < < N, positive definite.It is well known that the open-loop Nash control must satisfy [12]: u i= -RIB@i; = -Qi x-AT@i, @i(tf)=Kifxf, l<i<N (1. 3)   where qi is the costate vector associated with player "i'.When the transformation i Kix, is N is given by introduced, for 1 g g N, the open-loop Nash strategy (u )i and q)(t,0) is the system's transition matrix satisfying N '(t,O) (A- sjgj)@(t,O); @(t,t) I (1.7) j=l Note that the matrices Rij for j do not appear in the necessary conditions due to the fact that under open-loop strategy assumptions, each player optimizes his criterion knowing that Oui/Ox O, for _< _< N.For the open-loop Nash strategy and under the existence of a solution of the coupled Riccati system, the optimization problem has only one solution [2].The solution of system (1.5) is generally difficult to obtain due to the permanent coupling between the players' strategies.In [4] a series solution of system (1.5) is proposed but the coefficient are obtained solving several linear matrix equations.In [14] a numerical algorithm for the integration of (1.5) is given.A singular perturbation method for solving (1.5) is proposed in [10].For the case N 2, an iterative algorithm for solving (1.5) is given in [5].For the case N 2 and Q2-cQ1, where is a scalar, an analytic solution of system (1.5) was pointed out in [1].In this paper we obtain an explicit solution of system (1.5) for a case more general than the one proposed in [1].Also, a different type condition expressed in terms of the existence of certain coupled algebraic Riccati matrix system is proposed.
2. ANALYTIC SOLUTION OF COUPLED RICCATI SYSTEM BASED ON ALGEBRAIC TRANSFORMATIONS.
For convenience, the necessary conditions to be satisfied (1.1), (1.3), are rewritten in a matrix form for appropriate matri-ces L2,L3,...,LN in R nn to be determined Thus problem (2.1), (2.2) is equivalent to the following one The purpose of this transformation is to find under what conditions the player's optimization problem can be decoupled.In fact, note that if L2,L3,...,LN satisfy the system LjQ Qj, LjA T ATLj, 2 < j < N (2.6) the matrix T-IMT is reduced to a block triangular form and the costate vectors q/1,w2,...,wN, are coupled only via the terminal conditions (2.2).
Note that for the case N 2 and the matrices Q1 and Q2 are proportional, i.e., Q2 cQ1, for ome scalar c, taking L cl, one gets solutions of system (2.6) for N 2. Thus the case studied in [1] is a particular case of (2.6).
In order to characterize the existence of solutions for the algebraic system (2.6), we recall the concept of tensor product of matrices.If A,B are matrices in R m x n and R k x s, respectively, then the tensor product of A and B, denoted A (R) B, is defined as the partitioned matrix If A E R rax n, we denote A(R)B= allB al2B alnB amlB am2B amnB alj A.j= l < j < n; vecM= If M,N and P are matrices of suitable dimensions, then using the column lemma [7, p. 410], we get vec(MNP) (pT (R) M) vecN   (2.7) Taking into account (2.7), if we apply tensor products in each equation of system (2.6), it may be written in the equivalent form where 2 < j < N and C vecLj vec[0,Qj] (2.8) I(R)AT_A(R) I 1 C (2.9) QI(R)I If we denote by C + the Moore-Penrose pseudoinverse of C, then from Theorem 2.3.2 of [11, p.24], the system (2.9) is compatible if and only if CC + vec[0,Q/] vec[0,Q/] (2.10) Furthermore, under condition (2.10), the general solution of (2.9) for Lj is given by vecL/= C + vec[0,Q2 + (I -C + C)z where I denotes the identity matrix in Rn2x and z is an arbitrary vector in Rn2.We recall that C + may be computed by using MATLAB [8].
Let us assume the existence of solutions Lj of (2.8) for 2 < j g N, then from (2.4), (2.5)  (2.20) Note that [I,0]G(0)= I, and from the continuity of G, there exists an interval 0 < s < Now, from (2.3) and (2.13), it follows that ,I,(s) L ,r() +,(), 2 _< j _< g and from (2.15)and (2.19) From (2.13) and the relations qj(t)= Kj(t)z(t), 1 < j < N, it follows that where G is defined by (2.18).Thus the following result has been proved: THEOREM 1.Let us assume that matrices A and Qi for 1 _< _< N, satisfy the condition (2.9 where C is defined by (2.8), and let Lj be the solution of (2.8) for 2 < j _< N. Then there exists a positive number 6 such that on the interval [t1-6, tl] the unique solution of the coupled Riccati system (1.5)is given by (2.23), (2.24).
REMARK 1.Note that the case N 2, Q2 aQ1, where a is a scalar, is a particular case of the previous theorem taking L 2 aI.It is important to note that from (2.23) and (2.24), we have the following relation between K1 (t) and K2(t): K2(t L2KI(t +exp(AT(tft))[K2l-L2gly]{[I,O]G(ty-t)} -1 and since the function {[I,O]G(tl-t)}-I is involved in the computation of gl(t), the computational cost is reduced because K2(t is expressed in terms of Kl(t).Finally we recall that efficient methods for computing matrix exponentials and integrals involving them and that appear in the expression of G(s), may be found in [13].These procedures are extremely easy to implement and yield an estimation of the approximation error.Riccati type matrix equations with rectangular coefficients arise for instance in the problem of finding a state estimate feedback controller [3] and in the transformation of ill-conditioned linear systems to a block diagonal form [6,9].An efficient method to find solutions of such equations may be found in [15].The aim of this section is to propose another way to find an analytic solution of the coupled Riccati differential system (1.5).Note that system (1.5) may be written in the following compact form where and K'(t) -Q-K(t)A BK(t) + K(t)SK(t); K(tI) K! where V(t) E C n x, and Z(t) C Nn x n.If we define the matrix function exp((t-s)A) I exp((t-v)A)Sexp((v-z)B)dv S(t,s) ti (3.9) 0 exp((t-s)Bo) then an easy computation yields Ao -S] (o/ot)s(t,) s(t,) 0 Bo (3.10) and thus S(t,s) is a fundamental matrix of (3.8) and the unique local solution of (3.8) in a neighborhood of I is given by S(tI, tf) I(N + 1)n Note that V(![)= In and thus in a neighbourhood Y containing I, V(t) is invertible.Now, let us define the CnXn-values matrix function U(t)= Z(t)(V(t)) -x for teJ.Note that from (3.8)it follows that V'(t) AoV(t)-SZ(t) and Z'(t) BoZ(t) Computing it follows that U'(t) Z'(t)(V(t)) -1 Z(t)(V(t)) -1 V'(t)(V(t)) -1 BoU(t) U(t)Ao + U(t)SU(t) for all ( J. Hence K(t)= U(t)+ Xo is the solution of (3.1), defined on the interval J. From (3.10) it follows that V(t) exp((t tl)Ao)-| exp((t v)Ao)Sexp((v tl)Bo)dvU !exp((t-tl)AotI-.exp((t!v)Ao)Sexp(vBo)dv exp( g(t) exp((tti)Bo)U !-1 K(t) Xo + exp((t ti)Bo)U!I .exp((tIv)Ao)Sexp(vBo)dv exp( exp( f t)Ao)) tl (3.11) hu if the gebre meeati muation (3.4) h solution Xo, hen the oution K(t) of (3.1)   defined by (3.11) where UI, Ao d Bo e given by (3.7). he solution is defined in the eighh o[ t] where V(t) is ivertible.
ACKNOWLEDGEMENt.This pnr h n ptily supported by the D.G.I.C.Y.T. PS87-0064 d the NAO grit 900040.

3 .
ANALYTIC SOLUTION OF COUPLED RICCATI DIFFERENTIAL SYSTEM BASED ON THE EXISTENCE OF A SOLUTION OF A COUPLED ALGEBRAIC RICCATI SYSTEM.
S,S2,...,SN]Let us assume that the rectangular algebraic Riccati equation -Q-XA-BX+XSX=O U.f K I X o admits a solution X o 6_ C Nn x n and let us consider the change V(t) K(t) X o Then problem (3.1) is equivalent to the following one U'(t) BoV(t) U(t)Ao + U(t)SV(